Energy Bands & Current Carriers in Semiconductors (PDF)

# Energy Bands & Current Carriers In Semiconductors ## Chapter at a Glance - **Introduction:** A crystal is a systematic arrangement of atoms. A three-dimensional lattice is defined by the vectors. A pure crystal is constructed in such a way that it maintains directional invariance. The lattice i...

# Energy Bands & Current Carriers In Semiconductors ## Chapter at a Glance - **Introduction:** A crystal is a systematic arrangement of atoms. A three-dimensional lattice is defined by the vectors. A pure crystal is constructed in such a way that it maintains directional invariance. The lattice is a periodic point in space and is a mathematical abstraction. - **Energy Bands Theory in Crystals:** The total energy *E* of a conduction electron is given by: $E = \frac{h^2k^2}{2m}$ This indicates the parabolic dependence between the energy and wave vector as shown in the Figure for constant effective mass *m*. In this figure, - the horizontal line *Ec* indicates the edge of the conduction band, - the horizontal line *Ev* is called the edge of the valence band, - the dotted horizontal line near *Ec* is called the donor level *No*, - the dotted horizontal line near *Ev* is called the acceptor level *NA*. The band gap *I* is defined as: $Eg = Ec - Ev$ ### Insulators In some crystalline solids, the forbidden energy gap is very large, both bands being parabolic in nature. In such solids, at ordinary temperatures only a few electrons can acquire enough thermal energy to move from the valence band into the conduction band. ### Metals A crystalline solid is called a metal if the uppermost energy band is partly filled or the uppermost filled band and the next unoccupied band overlap in energy. Metal has an interpenetrating band structure; in metal, the electrons in the uppermost band find neighbouring vacant states to move in, and thus behave as free particles. ### Semiconductors The crystalline material for which the width of the forbidden energy gap varies between metal and insulator is referred to as semiconductors. Germanium and silicon having forbidden gaps of 0.78 eV and 1.12 eV, respectively, at 0K, are typical elemental semiconductors. ### Intrinsic & Extrinsic Semiconductor The electron and hole concentration in an intrinsic semiconductor are equal because carriers within a very pure material are created in pairs. Doped semiconductors whose properties are controlled by adding the impurity atoms are called extrinsic semiconductors. Doping increases the conductivity of a semiconductor. ### Carrier Drift Any motion of free carriers in a semiconductor leads to a current. This motion can be caused by an electric field due to an externally applied voltage, since the carriers are charged particles. We will refer to this transport mechanism as carrier drift. ### Diffusion The transportation of charge carriers in semiconductors may be accounted for by a mechanism called diffusion. ## Multiple Choice Type Questions 1. Intrinsic carrier concentration of a given semiconductor depends on - a) Bandgap - b) temperature - c) Bandgap and temperature - d) none of these **Answer:** (c) 2. Diffusion current in semiconductor flows due to - a) concentration gradient of carrier - b) electric field - c) both concentration & electric field current - d) none of these **Answer:** (a) 3. If temperature increases from very low value to high, then electron mobility - a) decreases - b) increases - c) increases then decreases - d) remains constant **Answer:** (c) 4. Electric field increases from very low value to high value then carrier velocity - a) increases - b) decreases - c) increases then saturate - d) decreases then saturate **Answer:** (c) 5. The electrical resistivity of a semiconductor is - a) about 10^4 ohm metre - b) in the range of 10^-6 - 10^-10 ohm metre - c) about 10^-3 ohm metre - d) less than 10^-26 ohm metre **Answer:** (a) 6. Which is the correct statement? - a) Effective mass *m** is positive when *E(k)* is concave up - b) Effective mass *m** is negative when *E(k)* is concave down - c) Effective mass *m** is infinite at points where the curve changes concave up to concave down or vice-versa (called "point of inflection") - d) All of these **Answer:** (d) 7. In GaAs when the electron rises from central valley to satellite valley, the effective mass of the electron becomes - a) less - b) more - c) zero - d) infinity **Answer:** (b) 8. One plane intercepts axis at 1, ∞, ∞. Miller indices of that plane is - a) {111} - b) (1∞ ∞) - c) (100) - d) [100] **Answer:** (d) 9. Which metal is suitable for ohmic contact with p-type silicon? - a) Fe - b) Cu - c) Al - d) Au **Answer:** (c) 10. Electron transition in direct band gap semiconductor involves - a) a change of momentum of electron - b) Dependence on band gap - c) No change of momentum for electron - d) None of these **Answer:** (b) 11. Electron effective mass depends on - a) Curvature of band - b) band gap - c) Doping concentration - d) temperature **Answer:** (a) 12. At 0K semiconductor has - a) Empty valence band and filled conduction band - b) Filled valence band and empty conduction band - c) Partially filled valence and conduction band - d) Holes in valence band **Answer:** (b) 13. In a degenerate n-type semiconductor Fermi level lies - a) Inside the conduction band - b) near the valence band - c) near the conduction band - d) at the middle of forbidden band **Answer:** (a) 14. Si has the lattice patterns of - a) FCC type - b) Hexagonal type - c) Diamond type - d) Zinc blende type **Answer:** (c) 15. Doping effect of semiconductor results with the change of - a) Fermi level only - b) Bandgap only - c) Electrical conductivity only - d) all of these **Answer:** (d) 16. The doping level of emitter region of a transistor is - a) greater than collector and base regions - b) less than collector and base regions - c) less than base region but greater than collector region - d) greater than base region but less than collector region **Answer:** (a) 17. When a positive voltage is applied to an n-type semiconductor with respect to the metal, the barrier between the semiconductor with respect to the metal, the barrier between the semiconductor and metal - a) increases - b) decreases - c) remains same - d) none of these **Answer:** (b) 18. GaAs is preferred to Si for high temperature operation of semi-conductor device because GaAs - a) is direct band gap in nature - b) possesses higher energy band gap - c) is a compound semi-conductor - d) possesses smaller carrier effective mass **Answer:** (c) 19. The basic lattice structure of silicon is - a) simple cubic - b) edge-centered cubic - c) face-centered cubic - d) body-centered cubic **Answer:** (c) 20. At T=0K, the Fermi-Dirac distribution function vs energy plot takes the form - a) step - b) linear - c) parabolic - d) exponential **Answer:** (a) 21. A p-type semiconductor contains holes and - a) Positive ions - b) Negative ions - c) Acceptor atoms - d) Donor atoms **Answer:** (c) 22. Diffusion of free electrons across the junction of an unbiased diode produces - a) Forward bias - b) Reverse bias - c) Breakdown - d) the Depletion Layer **Answer:** (d) 23. When a pentavalent impurity is added a semiconductor becomes - a) Positively charged - b) Negatively charged - c) Neutral - d) None of these **Answer:** (c) 24. Under high electric fields, in a semiconductor with increasing electric field - a) The mobility of charge carriers decreases - b) The mobility of charge carriers increases - c) Velocity of carriers saturate - d) both (a) and (c) **Answer:** (d) 25. The probability of recombination of EHP in semiconductor is proportional to - a) density of electrons - b) density of holes and electrons - c) density of holes - d) none of these **Answer:** (b) 26. Diffusion constant of holes and electrons are in ratio 4:1. Then the mobility of holes and electrons will be in the ratio - a) 4:1 - b) 16:1 - c) 1:4 - d) 1:16 **Answer:** (a) 27. A semiconductor which behaves like an insulator at zero Kelvin is called - a) intrinsic semiconductor - b) extrinsic semiconductor - c) elemental semiconductor - d) degenerate semiconductor **Answer:** (c) 28. In a semiconductor the hole diffusion length *L* is given by - a) *D*τ - b) (*D*τ)^1/2 - c) *D*/τp - d) (*D*τ)^1/2 **Answer:** (d) 29. Hall voltage is proportional to - a) velocity - b) magnetic field - c) both (a) and (b) parallel to velocity - d) both (a) and (b) perpendicular to velocity **Answer:** (b) 30. When a positive voltage is applied to a p-n junction structure the barrier potential - a) increases - b) decreases - c) remains same - d) none of these **Answer:** (b) 31. Electron transition in in-direct band gap semiconductor involves - a) a change of momentum of electron - b) dependence on band gap - c) no change of momentum of electron - d) none of these **Answer:** (b) 32. Effective electron mass depends on - a) curvature of band - b) band gap - c) doping concentration - d) temperature **Answer:** (a) 33. Metal n-type semiconductor form ohmic contact if - a) *m* > *sn* - b) *m* = *sn* - c) *m* < *sn* - d) none of these **Answer:** (a) 34. Energy bandgap of GaAs at 0 K is - a) 1.12 eV - b) 0.66 eV - c) 1.43 eV - d) 3.40 eV **Answer:** (d) 35. According to the E-k diagram, Si is - a) direct bandgap - b) indirect bandgap - c) both (a) and (b) - d) none of these **Answer:** (b) 36. Boltzmann approximation is valid for - a) higher energy states - b) lower energy states - c) both (a) and (b) - d) None of these **Answer:** (d) 37. Under forward bias, p – n junction current flows mainly due to - a) diffusion - b) drift - c) both (a) and (b) - d) none of these **Answer:** (c) 38. Intrinsic Fermi level (EF₁) will be slightly above the midgap energy level (Emidgap), if - a) *m*> *m* - b) *m* < *m* - c) *m* = *m* - d) any one of these **Answer:** (d) 39. Solar cell is a - a) photodetector - b) photodiode - c) photovoltaic device - d) optical emitter **Answer:** (c) 40. If *V* is the voltage applied to the metal with respect to the p-type semiconductor in a MOS capacitor then *V*< 0 corresponds to - a) Depletion - b) Accumulation - c) Inversion - d) Strong inversion **Answer:** (a) 41. The unit of Density of State Function is - a) number/unit energy/ unit volume - b) energy / unit volume - c) energy/ unit area - d) None of these **Answer:** (a) 42. K-space diagram in a crystal is a plot of - a) electron density versus momentum - b) electron energy versus momentum - c) quantum number versus momentum - d) electron energy versus density of energy states **Answer:** (b) 43. Intrinsic carrier concentration is given by - a) *n* = *n*/ *po* - b) *n* = *po*/ *no* - c) *n* = ( *no* * po* )^1/2 - d) *n* = *no* * po* **Answer:** (c) 44. Conductivity is defined as the ratio of - a) drift current density to electric field - b) drift current density to practical density gradient - c) particle flux to particle density gradient - d) drift velocity to electric field **Answer:** (a) 45. Mobility is a parameter which relates - a) drift current density to electric field - b) carrier drift velocity to electric field - c) resistance to current - d) mobility to diffusion coefficient **Answer:** (b) ## Short Answer Type Questions 1. What is Miller indices of a crystal? A plane intercepts at 2a, b/2, 3c in a simple cubic unit cell. What are the Miller indices of the plane? **Answer:** Miller Indices are defined as the reciprocals of the fractional intercepts which the plane makes with the crystallographic axes. The method by which indices are determined is best shown by example. Recall, that there are three axes in crystallographic systems. Miller indices are represented by a set of 3 integer numbers. Reciprocal of the intercept will be 1/2, 2, and 1/3 . It means intercepts along x, y, and z axes are 1/2, 2, and 1/3, respectively. 2. Draw and explain E-K diagram for a direct and an indirect band gap semiconductor with suitable example. **Answer:** The total energy *E* of a conduction electron is given by: $ E= \frac{h^2k^2}{2m} $ This indicates the parabolic dependence between the energy and wave vector as shown in the Figure for constant effective mass *m*. In this figure, - the horizontal line *Ec* indicates the edge of the conduction band, - the horizontal line *Ev* is called the edge of the valence band, - the dotted horizontal line near *Ec* is called the donor level *No* (to be explained later on), - the dotted horizontal line near *Ev* is called the acceptor level *NA* (to be explained later on). The band gap *I* is defined as: $Eg = Ec - Ev$ The above first figure shows a direct band gap semiconductor as the minima of the CB and maxima of the valence band are at the same line. So the electrons from CB can jump to the valence band without any extra loss of energy. It is suitable for optoelectronic devices. Example GaAs. The above second figure shows an indirect band gap semiconductor as the minima of the CB and maxima of the valence band are not vertically at the same line. So the electrons from CB can jump to the valence band indirectly with extra loss of energy. It is not suitable for optoelectronic devices. Example Si, Ge. 3. a) What are mobility and conductivity? **Answer:** The mobility μ of a carrier in an operational sense is defined as the proportionality constant between the average drift velocity Vp of a (ensemble of) carriers in the presence of an electrical field E *VD = μ· Ε* Conductivity can be defined simply by Ohms Law. *V= IR* Where *R* is the resistance, *I* the current and *V* the voltage present in the material. Thus from this relationship conductivity is found. The conductivity depends on the number of charge carriers (number of electrons) in the material and their mobility. b) What are the effects of temperature and doping on mobility? **Answer:** Mobility increases for both increase of doping and increase in temperature. 4. Compare 'drift' and 'diffusion' transport in a semiconductor. **Answer:** Drift is, by definition, motion of charged particle with the application of an applied electric field. When an electric field is applied across a semiconductor, the carriers start moving, producing a current. The positively charged holes move with the electric field, whereas the negatively charged electrons move against the electric field. The motion of each carrier can be described as a constant drift velocity, va. This constant takes into consideration the collisions and setbacks each carrier has while moving from one place to another. It is considered a constant though, because the carriers will eventually go the direction they are supposed to go regardless of any setbacks, especially if you look at the direction of all the carriers, instead of each one individually. Drift current in a semiconductor is the resultant of carrier drift. Because we are talking about a semiconductor, or specific areas in a semiconductor, we are interested in the current density. When dealing with drift current, we are interested in the current density due to drift, and drift arises in response to an electric field. Drift current also depends on the ability of the carriers to move around in the semiconductor, or the electron and hole mobility. Another parameter drift current depends on the carrier concentration, because you have to have carriers in order for there to be current. Each one of these carriers has a charge, but in this case we will only take *q* as a magnitude. Diffusion is the process of particles distributing themselves from regions of high concentration to regions of low concentration. If this process is left unperturbed, there will eventually be a uniform distribution of particles. Diffusion does not need external forces to act upon a group of particles. The particles move about using only thermal motion. If we let the particles be carriers, so as they move around they take charge with them. The moving of charge will result in a current. We call this current due to diffusion. The diffusion current in a metal-semiconductor diode is derived based on the assumption that the depletion layer is large compared to the mean free path, so that the concepts of drift and diffusion are valid. We start from the expression for the total current and then integrate it over the width of the depletion region: $J₁=q(μ,ηε+Dan)\frac{dn}{dx}$ The first part of the current density equation is for drift and second part is for diffusion. 5. A Si sample is doped with 10^17 as atoms/cm³ (n = 1.5 ×10^10). What is the equilibrium hole concentration *po* at 300 k? Determine the difference between Fermi level and intrinsic level. Draw the energy band diagram with proper labels. **Answer:** *po* *no* = *n*² *10^17* *po* = (1.5\*10^10)² *po* = 2.25\*10^20 / 10^17 = 2.25\*10^3 #### The energy band diagram Total energy *E* of conduction electron is given by: $E = \frac{h^2k^2}{2m}$ The parabolic dependence between the energy and wave vector is shown in the Figure for constant effective mass *m*. The band gap *Eg* is defined as *Eg = Ec - Ev*. Where *Ec* = Conduction band edge *Ev* = Valence band edge *Eg* = Band gap *Ef* is the energy of the Fermi-level which is exactly at the centre of the forbidden energy gap in the case of intrinsic semiconductor. i.e., EF = ( *Ec + Ev *) / 2 6. Qualitatively discuss the variation carrier concentration with temperature for extrinsic Si with the help of proper diagram. **Answer:** The Fermi-Dirac statistics expresses the probability with which the electron will occupy the energy level *E*. $ f(E)=\frac{1}{1+e^{(E-EF)/kT}} $ Where *k* is Boltzmann's constant and *Ef* is Fermi level of energy. The function *f(E)*, the Fermi-Dirac distribution function gives the probability that an available energy state E will be occupied by an electron at absolute temperature T. The plot of *f(E)* as a function of *E* is given in figure. #### Case 1 If T > 0K, If E = EF, f (E) = 1/2 i.e. the probability of occupancy of electron is ½ when energy becomes equal to Fermi energy. For T→ 0, If E = EF, f (EF) indicates the transition point. 7. Derive the steady state diffusion equation for holes in Si. What do you mean by diffusion length? **Answer:** The transport of charge carriers in semiconductors may be accounted for by a mechanism called diffusion. Diffusion current is the net flow of the randomly moving electrons and holes from a region of high carrier concentration to regions of lower carrier density. It is analogous to Fick's law of classical thermodynamics. For electrons, the diffusion-current density is given by: $ J = eD\nabla Vn$ Where *Vn* is the gradient of electron concentration and *D* is the electron diffusion constant. The total current due to the motion of holes by drift and diffusion is: *J, = e(µ, PoE-D, Vp)* Diffusion length means the depth of diffusion of impurity carriers doped within the semiconductor material. 8. Sketch the ideal energy band diagram of metal-semiconductor junction when QM < S. Explain why this is ohmic contact. **Answer:** Ideal metal-semiconductor contacts are ohmic when the charge induced in the semiconductor in aligning the Fermi levels is provided by majority carries. For example, in the ∅m < ∅s, (n-type) case of Fig: a, the Fermi levels are aligned at equilibrium by transferring electrons from the metal to the semiconductor. This raises the semiconductor electron energies (lowers the electrostatic potential) relative to the metal at equilibrium (Fig b). In this case the barriers to electron flow between the metal and the semiconductor is small and easily overcome by small voltage. 9. What do you mean by effective mass? Derive the expression of effective mass. How can effective mass differ from actual mass and in which condition effective mass will be positive, negative and infinity? **Answer:** The concept of mass of the carriers is extremely important in solid-state electronics. This mass is different from that of free carrier mass and the free carrier mass needs to be replaced by the effective mass to account for the effects of crystalline force. The effective carrier mass along a particular direction (m*) is given below: $m^* = \frac{momentum (p) along this direction}{velocity (v) along the same direction}$ We can write, $p = ħ k$ The term ħ / (2π) is called ħ and is called the normalized Planck’s constant or the Dirac’s constant and the term *k* = (2π) / λ is known as the carrier wave vector ( ħ ). Therefore the equation can be expressed as, $ pk = ħ k$ The velocity as written in equation must be the group velocity $\frac{δω}{dk}$ where the frequency ω=E/ħ in which E is the total energy of the carrier and not at all the phase velocity. Therefore the velocity of the carrier is $\frac{δω}{dk}$. Thus the mass of the carrier should, in general, be written as. $m* = ħ^2 \frac{d^2E}{dk²}$ Therefore the effective mass of the carriers can be expressed as, $m = ħ^2 \frac{d^2E}{dk²}$ From the above equation. We observed that effective mass changes with the slope of the E-k curve. This E-k relation is called the dispersion relation which changes from semiconductor to semiconductor, and consequently the m* also changes. Thus, mass can be a function of energy and changes with external physical conditions. Incidentally, from Newton’s second law we can prove that the acceleration effective mass $m_a = ħ^2 \frac{d^2E}{dk²}$ #### Derivation: From Newton's second law, we can write the force F on the carrier is given by: $ F = \frac{dp}{dt} = \frac{d(ħ k)}{dt}= ħ\frac{dk}{dt} $. Since *p* = *ħk* Also *F* can be described as F = *m*a, where *a* is acceleration of the carrier. Thus *a* = $\frac{1}{m} \frac{dF}{dt}$ = $\frac{1}{ħ} \frac{d(dE)}{dt}$ We know that $v_g = \frac{dE}{ħ dk}$ is the group velocity. Combining the above three equations, we get: $m_a = ħ^2 \frac{d^2E}{dk²}$ The acceleration effective mass also called the curvature effective mass. These two definitions yield the same result when $Ec^2k i.e. E-k relation is parabolic. For any deviation from the parabolicity these two definitions of the effective mass will not converge to the same expression. The effective momentum mass of the carriers as given by equation affects all the properties of semiconductors such as electronic heat capacity, diffusivity to mobility ratio, the Hall co-efficient, all types of transport co-efficient and changes with electron concentration and other externally controllable parameters. In the expression of effective mass we have *d^2E/dk^2*. The curvature of the band determines the electron effective mass. The curvature of *d^2E/dk^2* is positive at conduction band minima, but is negative at the valence band maxima. Thus electrons near the top of the valence band have negative effective mass. Valence band electrons with negative charge and negative mass move in an electric field in the same direction as holes with positive charge and positive mass. 10. What are direct band gap and indirect band gap semiconductors? Draw the E-K diagrams for Si and GaAs. **Answer:** In physics of semiconductor device, a direct bandgap means that the minimum of the conduction band lies directly above the maximum of the valence band in momentum space. In a direct bandgap semiconductor, electrons at the conduction-band minimum can combine directly with holes at the valance band maximum, while conserving momentum. The energy of the recombination across the bandgap will be emitted in the form of a photon of light. This is radiative recombination which is also called spontaneous emission. In Indirect bandgap semiconductors such as crystalline silicon, the momentum of the conduction band minimum and valence band maximum are not the same, so a direct transition across the bandgap does not conserve momentum and is forbidden. Recombination occurs with the mediation of a third body, such as a phonon or a crystallographic defect, which allows for conservation of momentum. These recombinations will often release the bandgap energy as phonons, instead of photons, and thus do not emit light. Light emission from indirect semiconductors is very inefficient and weak. So many new techniques are there to improve light emission by indirect semiconductors. Prime example of a direct bandgap semiconductor is gallium arsenide - a material commonly used in laser diodes. Si is an indirect band gap semiconductors so it is not used and optoelectronic source. 11. What is am bipolar transport? Why carrier generation and recombination rates are equal in thermal equilibrium? **Answer:** A bipolar transport is a process in which electrons and holes diffuse, drift and recombine with some effective diffusion co-efficient, mobility and life time. In thermal equilibrium, concentration of electrons and holes in conduction and valence bands are time independent. Since the net carrier concentrations are independent of time in thermal equilibrium, the rate at which electrons and holes are generated at the rate at which they reconsise must be end. 12. Define mobility and write down its unit. Also give an equation that relates the mobility and diffusivity of carriers in a semi-conductor. What is the significance of the equation? **Answer:** Mobility = Drift velocity per unit electric field is called mobility. Its unit is, cm²/V-S For non-uniformly doped semiconductor, *J* = 0 = *e*n*μ*E + *eD* *dn*/ *dx* *J* = 0 = *e*h*N*a*(n)*E + *eD* *d*r*a*(n)*/ *dn* So it may be written, 0 = -*e*h*N*a*(n)* *KT/C* (1/ *N*a *(n))* *d*N*a*(n)/*dn* + *eD* *d*N*a*( n )*/ *dn* = -e = μn The above equations is valid for the condition, *D* / μ = *KT/C*. In semiconductor, hole current must also be zero. So *Dp*/ μp = *KT/C*. So finally it may be written as, *D*₁/ μ₁ = *Dp*/ μp = *KT/e*. This relation is known as Einstein relation.. 13. Derive relationship between energy & momentum. **Answer:**: We know the total energy of a particle E = K. E + rest energy E = *mc*² = *mc*² / √1 + *v*²/*c*² When m is the mass; *m*o is the rest mass, *c* is the velocity of light, *v* is the velocity of the particle. Momentum varies with velocity as, *p* = *mv* = *m*o*v* /√1 - *v*²/ *c*² Again, *E*² = *mc*² = *mc*² / (1 - *v*²/ *c*²) so, *m*² *c*⁴ (1 - *v*²/ *c*²) = *mc*² or, *m*² *c*⁴ - *m*² *v*² *c*² = *mc*² or, *m*² *c*⁴ = *E*²(1 - *v*²/ *c*²) = *mc*² + *m*² *c*² *v*² again momentum *p* = *mv* so, *E* = *mc*² + *c*² *p*² is *p* is very small then, *E* ≈ *mc*² + *p*²(2*m*o) 14. a) What is density of states? **Answer:** In solid-state and condensed matter physics, the density of states of a system describes the number of states per interval of energy at each energy level that are available to be occupied by electrons. b) Explain the plot of Fermi-Dirac distribution function with energy for different temperatures. **Answer:** Refer to question 6 of Short Answer Type Questions. c) 3 Volts is applied across a 1-cm long Si bar. Determine mobility when the drift velocity is 10 cm/sec. **Answer:** μ = *va*/ E = 10*(cm/sec)/3 (v/cm) = 3.33\*10^3 *cm*²/*v*-s 5. What is the mass action law for the carrier concentration in a semiconductor? Write down the mathematical expression for Fermi Dirac probability function *f(E))* and plot *f(E)* VS *E/Ef* for three different temperatures: T=0K, 300K, 2000K and explain it. **Answer:** Theoretical analysis leads to the result that under thermal equilibrium, the product of free negative and positive concentrations is a constant independent of the amount of donor and acceptor impurity doping. It is given by *np = n*i² where *n*i is temperature dependent. Electrons are indistinguishable and identical particles with half-integer spin and obey the Pauli’s exclusion principle. The energy distribution of electrons in a solid is governed by Fermi-Dirac statistics. The probability of occupying any electronic state E by an electron is given by Fermi-Dirac distribution function as: $ f(E)=\frac{1}{1+ exp ^{(E-Ef)/kT}} $ Where -*k* = Boltzmann’s constant -T= Absolute temperature -*Ef* = Reference energy, called Fermi level -The Fermi-Dirac distribution function is generally called the Fermi function. -Consider two cases at T = OK. -This shows that the distribution takes the simple rectangular form at T=OK as shown in Fig. below. 6. What is a hetero-junction? How many types of hetero-junctions are possible? Draw the band diagrams of each type of hetero-junction, considering straddling. **Answer:** A heterojunction is the interface that occurs between two layers or regions of dissimilar crystalline semiconductors. These semiconducting materials have unequal band gaps as opposed to a homojunction. It is often advantageous to engineer the electronic energy bands in many solid state device applications including semiconductor lasers, solar cells and transistors ("heterotransistors") to name a few. The combination of multiple heterojunctions together in a device is called an heterostructure although the two terms are commonly used interchangeably. The requirement that each material be a semiconductor with unequal band gaps is somewhat loose especially on small length scales where electronic properties depend on spatial properties. A more modern definition of heterojunction is the interface between any two solid-state materials, including crystalline and amorphous structures of metallic, insulating, fast ion conductor and semiconducting materials. The three types of semiconductor heterojunctions organized by band alignment. Band diagram for straddling gap, n-n semiconductor heterojunction at equilibrium. The behaviour of a semiconductor junction depends crucially on the alignment of the energy bands at the interface. Semiconductor interfaces can be organized into three types of heterojunctions: (1) straddling gap (type I), (2) staggered gap (type II) (3) broken gap (type III) as seen in the figure. Away from the junction, the band bending can be computed based on the usual procedure of solving Poisson's equation. Various models exist to predict the band alignment. The simplest and least accurate model is Anderson's rule, which predicts the band alignment based on the properties of vacuum-semiconductor interfaces in particular the vacuum electron affinity. The main limitation is its neglect of chemical bonding. A common anion rule was proposed which guesses that since the valence band is related to anionic states, materials with the same anions should have very small valence band offsets. This however did not explain the data but is related to the trend that two materials with different anions tend to have larger valence band offsets than conduction band offsets. Tersoff proposed a gap state model based on more familiar metal-semiconductor junctions where the conduction band offset is given by the difference in Schottky barrier height. This model includes a dipole layer at the interface between the two semiconductors, which arises from electron tunneling from the conduction band of one material into the gap of the other. This model agrees well with systems where both materials are closely lattice matched such as GaAs/AlGaAs. The typical method for measuring band offsets is by calculating them from measuring exciton energies in the luminescence spectra. 17. 3 volt is applied across a 1 cm long Si bar. Determine mobility with the drift velocity is 104 cm/s. **Answer:** 3V is applied across a 1 cm long Si bar. Calculate mobility with drift velocity is 104 cm/s. We know that, *V*₁ = *μ*E [where μ = mobility] *V* = Drift velocity, E = Electric field